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On the field of definition of superspecial polarized abelian varieties and type numbers

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On the field of definition of superspecial polarized abelian varieties and type numbers COMPOSITIO MATHEMATICA TOMOYOSHI IBUKIYAMA TOSHIYUKIKATSURA On the field of definition of superspecial polarized abelian varieties and type numbers Compositio Mathematica, tome 91, no 1 (1994), p. 37-46. <http://www.numdam.org/item?id=CM_1994__91_1_37_0> © Foundation Compositio Mathematica, 1994, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 37 On the field of definition of superspecial polarized abelian varieties and type numbers TOMOYOSHI IBUKIYAMA1 and TOSHIYUKI KATSURA2 1 Department of Mathematics, College of General Education, Osaka University, Toyonaka 560, Japan 2Department of Mathematical Sciences, The University of Tokyo, Tokyo 153, Japan Received 17 August 1990; accepted in final form 12 January 1993 Compositio Mathematica 91: 37-46, 1994. C) 1994 Kluwer Academic Publishers. Printed in the Netherlands. Many algebro-geometrical phenomena on an abelian variety A which is the product of supersingular elliptic curves over the field of characteristic p &#x3E; 0 are related to the arithmetic theory on quaternion hermitian lattices, as was shown, for example, in [3], [10], [11], [8]. Now, in this paper, we shall study the relation between the field of definition of principally polarized abelian varieties (A, 0), where A is as above, and some arithmetic nature on quaternion hermitian lattices, for example, Hecke operators, type numbers and so on. When dim A = 1, that is, when A is a supersingular ellipt

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